Electricity and Magnetism 65 Chapter 7 ELECTROMAGNETIC WAVES 7.1 Maxwell’s Equations 1) Maxwell’s Equations Gauss’law for electricity ∫ surfaceclosed A.dE r r = o q ε (7.1) Gauss’law for magnetism ∫ surfaceclosed A.dB r r = 0 (7.2) Faraday ‘s law ∫ pathclosed s.dE r r = - t B ∂ Φ∂ (7.3) Ampere-Maxwell Law ∫ pathclosed s.dB r r = t εµ E oo ∂ Φ∂ + µ o i (7.4) 2) Vector calculus (or vector analysis) del operator : z i y i x i zyx ∂ ∂ + ∂ ∂ + ∂ ∂ =∇ r r r (7.5) • gradient : grad(V) = z V i y V i x V iV zyx ∂ ∂ + ∂ ∂ + ∂ ∂ =∇ r r r (7.6) Maps scalar fields to vector fields. Measures the rate and direction of change in a scalar field. • divergence : div( F r ) = ( ) zzyyxxzyx FiFiFi. z i y i x iF rrrrrr r ++         ∂ ∂ + ∂ ∂ + ∂ ∂ =∇ = z F y F x F z y x ∂ ∂ + ∂ ∂ + ∂ ∂ (7.7) Maps vector fields to scalar fields. Measures the magnitude of a source or sink at a given point in a vector field. Property : ∫ ∫ =∇ surfaceclosedvolume A.dF.dVF r r r (7.8) • curl: curl( F r ) = rot( F r ) = ( ) zzyyxx x zyx x FiFiFi z i y i x iF rrrrrr r ++         ∂ ∂ + ∂ ∂ + ∂ ∂ =∇ Electricity and Magnetism 66 = zyx zyx FFF zyx iii ∂ ∂ ∂ ∂ ∂ ∂ r r r (7.9) Maps vector fields to vector fields. Measures the tendency to rotate about a point in a vector field. Property : ∫ ∫ =∇ pathclosedsurface s.dFA.dF x r r r r (7.10) • Laplacian: ))div(grad(VV VV 2 =∇∇=∇=∆ = V z i y i x i. z i y i x i zyxzyx         ∂ ∂ + ∂ ∂ + ∂ ∂         ∂ ∂ + ∂ ∂ + ∂ ∂ rrrrrr = 2 2 2 2 2 2 z V y V x V ∂ ∂ + ∂ ∂ + ∂ ∂ (7.11) Maps scalar fields to scalar fields. A composition of the divergence and gradient operations. F r ∆ = F zyx 2 2 2 2 2 2 r         ∂ ∂ + ∂ ∂ + ∂ ∂ = zzyyxx iFiFiF r r r ∆+∆+∆ (7.12) Property : F r ∆ = F 2 r ∇ - ∇ x ∇ x F r (7.13) 3) Maxwell’s Equations in term of del operator Gauss’law for electricity ∫ ∇ volume .dVE r = ∫ surfaceclosed A.dE r r = o q ε = ∫ volume .dV ε ρ o It follows that E r ∇ = o ε ρ (7.14) Gauss’law for magnetism ∫ ∇ volume .dVB r = ∫ surfaceclosed A.dB r r = 0 It follows that B r ∇ = 0 (7.15) Faraday ‘s law ∫ ∇ surface A.dE x r r = ∫ pathclosed s.dE r r = - t B ∂ Φ∂ = - Ad. t B r r ∫ ∂ ∂ surface Electricity and Magnetism 67 It follows that E x r ∇ = - t B ∂ ∂ r (7.16) Ampere-Maxwell Law ∫ ∇ surface A.dB x r r = ∫ pathclosed s.dB r r = dt d E oo Φ εµ + µ o i = ∫         + ∂ ∂ surface AdJµ t E εµ ooo r r r = It follows that B x r ∇ = Jµ t E εµ ooo r r + ∂ ∂ (7.17) 4) Wave equation Applying (7.13) yields E r ∆ = E 2 r ∇ - ∇ x ∇ x E s (7.18) With J = 0 and ρ = 0 E r ∇ = 0 and ∇ x E s = - t B ∂ ∂ r (7.19) We have E r ∆ = ∇ x t B ∂ ∂ r = t ∂ ∂ B x r ∇ = t E εµ 2 2 oo ∂ ∂ r (7.20) Inserting (7.19) and (7.20) into (7.18) we have the wave equation E r ∆ - t E εµ 2 2 oo ∂ ∂ r = 0 (7.21) 7.2 Electromagnetic Waves An electromagnetic wave consists of oscillating electric and magnetic fields. The various possible frequencies of electromagnetic waves form a spectrum, a small part of which is visible light. An electromagnetic wave traveling along an x axis has an electric field E r and a magnetic field B r with magnitudes that depend on x and t E = E m sin(kx- ω t) (7.22) B = B m sin(kx- ω t) (7.23) where ω : angular frequency of the wave, k : angular wave number of the wave. These two components can not exist independently. The two fields continuously create each other via induction : the time varying magnetic field induces the electric field via Faraday ‘s law of induction, the time varying electric field induces the magnetic field via Maxwell ‘s law of induction. Electricity and Magnetism 68 a) b) Fig. 7.1 a) Electric field induced by magnetic field b) Magnetic field induced by electric field The key features of an electromagnetic wave - The electric field E r is always perpendicular to the magnetic field B r . The electric field E r and the magnetic field B r are always perpendicular to the direction in which the wave is travelling (the wave is a transverse wave). The cross product E r x B r always gives the direction in which the wave travels. - The fields always vary sinusoidally with the same frequency and in phase with each other. - All electromagnetic waves, including visible light, have the same speed c (3x10 8 m/s) in vacuum. The electromagnetic wave requires no medium for its travel. It can travel through a medium such as air or glass. It can also travel through vacuum. c k 1 B E oo m m = ω = εµ == B E (7.24) Fig. 7.2 : The electromagnetic spectrum 7.3 Energy Flow The rate per unit area at which energy is transported via an electromagnetic wave is given by the Poynting vector Electricity and Magnetism 69 o 1 S= E x B µ r r r (7.25) Fig. 7.3 : Electromagnetic wave The direction of S r (and thus of the wave’s travel and the energy transport) is perpendicular to the direction of both E r and B r . Since E r and B r are perpendicular o 2 o c EEB S µ = µ = = 2 m o E c µ sin 2 (kx-ωt) [W/m 2 ] (7.26) The time-averaged of S is called the intensity I of the wave 2 m o E I = 2c µ [W/m 2 ] (7.27) A point source of electromagnetic waves emits the wave isotropically (i.e. with equal intensity in all directions). The intensity of the waves at distance r from a point source of power P s is s 2 P I = 4 πr [W/m 2 ] (7.28) 7.4 Radiation pressure When a surface intercepts electromagnetic radiation, a force and a pressure are exerted on the surface. If the radiation is totally absorbed by the surface, the force is IA F= c (7.29) where I is the intensity of the radiation and A is the area of the surface perpendicular to the path of the radiation. If the radiation is totally reflected back along its original path, the force is 2IA F= c (7.30) The radiation pressure p r is the force per unit area r F p = A (7.31) Electricity and Magnetism 70 Problems 7.1) An electromagnetic wave with frequency 4x10 14 Hz travels through vacuum in the positive direction of an x axis. The wave has its electric field directed parallel to the y axis with amplitude E m . At time t = 0, the electric field at point P on the x axis has a value of E m /4 and is decreasing with time. What is the distance along the x axis from point P to the first point with E = 0 if we search in a) the negative direction of the x axis a) the positive direction of the x axis 7.2) An airplane flying at a distance of 10km from a radio transmitter receives a signal of intensity 10 µW/m 2 . What is the amplitude of the electric and magnetic component of the signal at the airplane ? If the transmitter radiates uniformly over a hemisphere, what is the transmission power ? 7.3) The maximum electric field 10m from an isotropic point source of light is 2V/m. What are the maximum value of the magnetic field and the average intensity of the light there ? What is the power of the source ? 7.4) Sunlight just outside earth’s atmosphere has an intensity of 1.4 kW/m 2 . Calculate the amplitude of the electric and magnetic field there, assuming it to be a plane wave. 7.5) A plane electromagnetic wave, with wave length 3m, travels in vacuum in the positive direction of an x axis. The electric field, of amplitude 300V/m, oscillates parallel to the y axis. What are the frequency, angular frequency and angular wave number of the wave ? What is the amplitude of the magnetic field component ? Parallel to which axis does the magnetic field oscillates ? What is the time-averaged rate of energy flow associated with this wave ? The wave uniformly illuminates a surface of area 2m 2 . If the surface totally absorbs the wave, what are the rate at which momentum is transferred to the surface and the radiation pressure on the surface ? 7.6) An isotropic point source emits light at wavelength 500nm, at rate of 200W. A light detector is positioned 400m from the source. What is the maximum rate B t ∂ ∂ at which the magnetic component of the light changes with time at the detector’s location ? 7.7) The basic equations of electromagnetism is called Maxwell’s equations which are given in the vacuum (J = 0, ρ = 0) as below: D ρ ∇ = r (Gauss’s law for magnetism) 0 B ∇ = ur (Gauss’s law for electricity) B E t ∂ ∇× = − ∂ ur ur (Faraday’s law) D H j t ∂ ∇× = + ∂ r r r (Ampere-Maxwell’s law) Where j E σ = r r , o D E εε = r r , HB o r r µµ = . Show that from Maxwell’s equation the following wave equation can be derived. 2 2 0 o o E E t ε µ ∂ ∆ − = ∂ r r Electricity and Magnetism 71 Homeworks 7 H7.1 A plane electromagnetic wave, with wave length λ [m], travels in vacuum in the positive direction of an x axis. The electric field, of amplitude E [V/m], oscillates parallel to the y axis. What are the frequency, angular frequency and angular wave number of the wave ? What is the amplitude of the magnetic field component ? Parallel to which axis does the magnetic field oscillates ? What is the time-averaged rate of energy flow associated with this wave ? The wave uniformly illuminates a surface of area 2m 2 . If the surface totally absorbs the wave, what are the rate at which momentum is transferred to the surface and the radiation pressure on the surface ? n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 λ 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 E 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 n 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 λ 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 E 25 50 75 100 150 200 250 300 350 400 450 500 550 600 650 700 n 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 λ 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 E 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 n 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 λ 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 4.5 5 5.5 6 6.5 E 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 Appendix I Factor Prefix Symbol Factor Prefix Symbol 10 24 yotta- Y 10 -24 yocto- y 10 21 zetta- Z 10 -21 zepto- z 10 18 exa- E 10 -18 atto- a 10 15 peta- P 10 -15 femto- f 10 12 tera- T 10 -12 pico- p 10 9 giga- G 10 -9 nano- n 10 6 mega- M 10 -6 micro- µ 10 3 kilo- k 10 -3 milli- m 10 2 hecto- h 10 -2 centi- c 10 1 deka- da 10 -1 deci d Appendix II Surface of a sphere of radius R : S = 4πR 2 Volume of a sphere of radius R : V = 4 πR 3 /3 Electricity and Magnetism 72 Circumference of a circle of radius R : C = 2πR ( ) ( ) 2/1 222 2/3 22 axa x ax dx + = + ∫ Appendix III Dot product of two vectors is a scalar A r • B r = | A r |.| B r |.cos(α) = A x B x + A y B y + A z B z Cross product of two vectors is a vector C r = A r x B r where | C r | = | A r |.| B r |.sin(α) and the direction of C r is determined by the right hand rule. The line integral of the vector F r along the curve L from A to B is a scalar ∫ • B A LdF rr = ∫ B A dLF )cos(|| α = ∫ x x B A xx dLF + ∫ y y B A yy dLF + ∫ z z B A zz dLF The surface integral of the vector F r through the surface A is a scalar ∫ • A dAnF r r = ∫ A dAF )cos(|| α = ( ) ∫ ++ A zzyyxx dAnFnFnF The volume integral of the scalar F over the volume V ∫ V FdV . n 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 λ 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 E 200 250 30 0 35 0 400 450 500 550 600 650 70 0 75 0 800 850 900 950 n 49 50 51 52 53. 70 0 75 0 800 850 n 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 λ 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 E 25 50 75 100 150 200 250 30 0 35 0 400 450 500 550 600 650 70 0 n 33 . yotta- Y 10 -2 4 yocto- y 10 21 zetta- Z 10 -2 1 zepto- z 10 18 exa- E 10 -1 8 atto- a 10 15 peta- P 10 -1 5 femto- f 10 12 tera- T 10 -1 2 pico- p 10 9 giga- G 10 -9